Elementary arithmetic, with brief notices of its history... by Robert Potts.

6
10. PROP. To determine tvhat must be the prime factors of the denominators of fractions which can be converted into terminating decimals.
respectively must be annexed to the respective numerators, and the equivalent
decimals will terminate and consist of one, two, three places respectively.
I T 10 10 5.
Thus1 - 'D.
2 2X10 2X10 10
3 3x100 300 30 75
4 4X100 4X100 100
5 5X1000 5000 625
' -g625.
8 8X000 8X1000 1000
In the first 100 natural numbers there are only 13 which are composed of factors
of 2 and of 5; it is obvious, therefore, that of the series of fractions whose denominators are the first 100 natural numbers (omitting unity) there will be 13 which
produce terminating decimals, and 86 repeating decimals. Hence it is clear that the
greater number of fractions when reduced will produce repeating decimals. In practice, however, repeating decimals are not generally employed, except in cases where
very minute accuracy is required in the result. In ordinary cases any approximate
degree of accuracy may be attained by considering repeating decimals to terminate at
any figure in the series of repeating figures, and the limit of error in excess or
defect can be definitely stated in each case. And the larger the number of decimal
figures be taken, the more nearly will the terminating decimal approximate to the
exact value of the repeating decimal,
If 3 be converted into its equivalent decimal,
1_ x o1000000 1oo000000 142857 _ 142857 1 1
7 7 x1000000 7 x1000000 1000000 1000000 1000000 7
it appears that 6 figures continually recur, and if 1, 2, 3, 4, 5, &c., figures be
taken successively as approximate values, it will appear how the successive
terminating decimals differ from the true value of the repeating decimal.
Here - = *142857142857142857..
Let '1, '14, '142, '1428, '14285, '142857, &c., be successively taken as approximate values of the recurring decimal.
If -1 be taken for a, then - is greater than '1, but less than *2.
~14 '14 '15..142 '142 '143.
~1428 '1428 '1429.
*14285 '14285 '14286.
*142857 '142857 '142858.
~1428571 '1428571 '1428572.
and so on. And the limits of error both in excess and defect from the true value can
be determined in every case.
Let the first assumption be taken: + greater than '1, but less than '2.
Then - = - = 13 error in defect.
7 7 10 70
*2 - 1 = error in excess.
7 10 7 70
Let the third be taken: a greater than '142, but less than '143.
- _ 142 _ _ 6 error in defect.
7 7 1000 7000
143 1 1 error in excess.
7 1000 7 7000